Neural Dynamic N-mixture Model

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.title[
#     Neural Dynamic N-mixture Model
]
.subtitle[
## A deep learning framework for inferring demographic rates from count data
]
.author[
### <strong>Speaker:</strong><br>François Leroy (he/him)<br><strong>Supervised by:</strong><br>Marta Jarzyna<br><br><br><br>
]
.institute[
### <br><br><br><br>The Ohio State University
]
.date[
### 2025-09-29
]

---

# Beyond abundance change

<div class="center" style="margin:0;">
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    Demographic rates
  </h2>
</div>

.pull-left[

* Changes in abundance are valuable, but demographic mechanisms offer deeper ecological insights

* Species' extinctions event are the result of increasing difference between birth and loss

* Demographic rates can be early warnings of species extinction

* Monitoring survival and recruitment can help anticipate species' extinction and understand better the current biodiversity crisis

]

.pull-right[

]

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# Beyond abundance change

<div class="center" style="margin:0;">
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    Problem
  </h2>
</div>

.pull-left[

* Data for demographic rates models are individual based (i.e. *individual encounter history data*)

* They are costly in resources and time

* Individual identification is not equal for all taxa

* Can be invasive/traumatic

* Limited spatial and temporal extent

]

.pull-right[

]

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# Beyond abundance change

<div class="center" style="margin:0;">
  <h2 style="margin:0 0 0em; line-height:1.05; display:inline-block;">
    Using abundance to infer demographic rates
  </h2>
</div>

.pull-left[

* However, abundance data are available at large spatial and temporal scale

* The idea of inferring birth/immigration (`$\mu$`) and death/emigration (`$\lambda$`) from experimental data is not new

* While overall change in abundance gives information about `$\lambda-\mu$`, the volatility of the time-series provide information about `$\lambda+\mu$` making `$\lambda$` and `$\mu$` identifiable

]

.pull-right[

<br>

]

.footnote[Wilkinson, 2011]

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# Dynamic N-mixture model

<div class="center" style="margin:0;">
  <h2 style="margin:0 0 0em; line-height:1.05; display:inline-block;">
    Hierarchical Model
  </h2>
</div>

**Observation process**

.pull-left[

`$$y_{i,j,t} \sim Binomial(N_{i,t}, p)$$`

]

.pull-right[

`$y =$` observed abundance<br>
`$p =$` detection probability

]

**State process**

.pull-left[

`$$N_{i,1} \sim Poisson(\lambda)$$`
<br>

`$$S_{i,t+1} \sim Binomial(N_{i,t}, \phi_{i,t})\\
R_{i,t+1} \sim Poisson(\gamma_{i,t})\\
N_{i,t+1} = S_{i,t+1} + R_{i,t+1}$$`

]

.pull-right[

`$\lambda =$` abundance at `$t = 1$`

<br>

`$\phi =$` survival probability<br>
`$\gamma =$` number of recruits
]

.footnote[Dail & Madsen, 2011]

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# Neural hierarchical model

.pull-left[

**Limitations of hierarchical framework**

* Linear or simple polynomial effects of covariates, even though true ecological responses are often unknown *a priori*

* MCMC algorithm scales poorly because inherently sequential, little possibility of parallelization

* Sensitive to priors and initial values

]

.pull-right[

**Limitations of Neural Networks**

* Usually lack inferential power, making it less relevant for ecological insights

* Doesn't distinguish between ecological process and imperfect detection

]

.center[

]

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# Neural hierarchical model

.center[

]

<br>

* Combines flexibility and scalability of neural networks with the inferential power of hierarchical models

* Output activation function according to the parameter to infer (sigmoid for probability, exponential for counts)

* Loss function: tailored from the model specific negative log-likelihood

* Combines decades of development in hierarchical modelling for ecological data with the flexibility and predictive power of Neural Networks

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# Neural Dynamic N-mixture model

.footnote[\* Krishnan et al. 2017]

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    Deep Markov Model*
  </h2>
</div>

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<br>

**State process**

`$$\begin{align*}
& N_{i,1} \sim Poisson(\lambda)
\end{align*}$$`

<br>

`$$\begin{align*}
& R_{i,t+1} \sim Poisson(\gamma_{i,t})\\
& S_{i,t+1} \sim Binomial(N_{i,t}, \phi_{i,t})\\
& N_{i,t+1} = S_{i,t+1} + R_{i,t+1}
\end{align*}$$`

**Observation process**

`$$\begin{align*}
& y_{i,j,t} \sim Binomial(N_{i,t}, p)
\end{align*}$$`

]

.pull-right-wide[

<br>

]

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# Example with the N-mixture model

* Show the likelihood

* You have to marginalize over n_max

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# The transition matrix

* As it it a Hidden Markov Model, the likelihood involves a transition between time `$t$` and `$t+1$`

* This is the bottleneck

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# Loss function of the DNMM

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# Optimization

* Hidden Markov Model `$\Rightarrow$` Deep Markov Model (Krishnan et al., 2017)

* Trade-off between memory use and speed

* I choose fast implementation with heavy memory use

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# Simulated data

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# Results of the simulation

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# Fitting on real data

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# Next step

* So far it is a simple MLP

* Promising to use a CNN